Synchronization and Nonlinear Integral Control

Indian Institute of Technology

Bombay

March 18, 2014 Roger Brockett

Harvard University

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What is to be explained and what tools will help?

Huygens Clocks

Today’s communication networks require precise synchronization. 2

Possible First Question

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Possible Second (Better) Question

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Properties of Huygens “Synchronization”

1.  Independent of coupling strength (unmodeled)

2.  Seems to equalize the originally unknown periods

3. Does not fix the relative amplitudes

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Uncertain Plant; Integral Control 1.  With system unknown, it fixes the steady state

value precisely.

2. Drives the error to zero but time constants can be large.

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Uncertain Plant; Integral Control

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Problem Statement

There are many such stabilization questions that find use in control some quite interesting from a mathematical point of view.

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A Stabilization Theorem: Background

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Background: The Schur-Horn Theorem

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In Pictures: Eigenvalues to Diagonals to Eigenvalues

In particular, by adding a skew-symmetric matrix we can make all eigenvalues real and equal. 11

What is involved in proving that any such diagonal can be realized?

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One approach to the proof is to look for the orthogonalmatrix ⇥ such that ||⇥TQ⇥�D|| is minimized and thenshow that the minimum is zero under the Schur-Hornconditions. Because there are many local minima, arrivingat a decisive conclusion requires a somewhat tedious secondderivative calculation.

A New Variation on the Schur-Horn Theorem (An eigenvalue placement with multiplicity)

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In Pictures: Eigenvalues to Diagonals to Eigenvalues

In particular, by adding a skew-symmetric matrix we can make all eigenvalues real and equal. 14

However if the eigenvalues are to be equal…..

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How to prove this?

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An important refinement

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How are we to use this fact?

What type of coupling can work? 18

Towards nonlinear integral control: what second order terms are available?

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Characterizing the resources in terms of the number of systems (clocks) and their degrees of freedom.

The exact terms do not offer interesting possibilities beyond those already present in x and dx/dt.

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Controllability with Linear and Quadratic Drift

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Controllability with Linear and Quadratic Drift

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Corrective Signal; the Lissajous figure in n-dimensions

When x is 2-dimensional and sinusoidal

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Corrective signal; relative phase

(

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Potential models for frequency equalization

Not Huygens-like

Strength of interaction not important

No integration

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A Numerical Example

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The Lissajous figure

Numerical Simulation

Showing numerical convergence to synchronization with phase offset

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Showing the transient response of the z variable.

Numerical Simulation

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Undamped oscillators coupled by an over damped back plane. Nonlinearities “rectify” out- of-phase oscillations producing a corrective signal.

Oscillator #1 Oscillator #2

Coupling through over damped backplane

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Is the solution truly periodic?

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To Disarm Potential Defenders of Huygens (1629 – 1695)

For reference: Newton 1642 – 1727 31

Roger Brockett, (2013) “Synchronization without Periodicity”, Festschrift in Honor of Uwe Helmke http://users.cecs.anu.edu.au/~trumpf/UH60Festschrift.pdf

Roger Brockett, (2013) “ Controllability with quadratic drift”, MATHEMATICAL CONTROL AND RELATED FIELDS Volume 3, Number 4, December 2013

For further details, see

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Abstract Version of the theorem on eigenvalue adjustment

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Conclusions We have given a result on eigenvalue placement for symmetric matrices perturbed by skew-symmetric matrices.

We have given an argument that Huygens synchronization involves a type of integral control, albeit a nonlinear form.

We have described how the “first bracket” controllable integrals can provide the necessary integral control.

We suggest that because averaging theory shows there is no periodic solution for the obvious model, whereas numerical simulation shows apparent synchronization, in fact, Huygens synchronization is not actually synchronization but highly confined near periodic irregular motion.

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